论文标题
在非convex-concave平滑的min-max问题中有效搜索一阶Nash平衡问题
Efficient Search of First-Order Nash Equilibria in Nonconvex-Concave Smooth Min-Max Problems
论文作者
论文摘要
我们提出了一种有效的算法，用于在$ \ min_ {x} \ max_ {y max_ {y} f（x，y）$的最小纳什均衡问题中找到一阶nash平衡问题，其中目标函数在变量中都很平稳，并且cove and cobave and cobave like y $ y $;套装$ x $和$ y $是凸面和“投影友好的”，$ y $紧凑。我们的目标是找到一个$（\ varepsilon_x，\ varepsilon_y）$ - 相对于平稳性标准，一阶Nash平衡比常用的近端梯度标准更强。所提出的方法非常简单：我们对原始函数执行近似近端迭代，而Nesterov的算法不精确地在正规函数上运行$ f（x_t，\ cdot）$，$ x_t $是当前的原始原始迭代。结果迭代复杂度为$ O（\ varepsilon_x {}^{ - 2} \ varepsilon_y {}^{ - 1/2}）$ to Googarithmic因子。作为副产品，选择$ \ varepsilon_y = o（\ varepsilon_x {}^2）$允许$ o（\ varepsilon_x {}^{ - 3}）$复杂性的复杂性，可以找到$ \ varepsilon_x $ - varepsilon_x $ - varepsilon_x $ - copationary-stationary-stationary-stationary-stationary-stationary-stationary primal erauau envel eenvel eenvel eenvel eenvel nevel nevel nevel nevel nevel formail forme nevel nevel formict。此外，当目标在$ y $方面强烈凹入时，我们算法的复杂性估计提高到$ o（\ varepsilon_x {}^{ - 2} { - 2} {κ_y}^{1/2} {1/2}）$，最高$κ__y$是条件编号，以调整条件编号。在这两种情况下，复杂性估计值是迄今为止最著名的，仅因（较弱的）近端梯度标准而闻名。同时，我们的方法是“用户友好的：”（i）该算法是在运行Nesterov加速算法的变体中构建的，并避免了外部步骤； （ii）收敛分析在具有不精确的甲骨文的加速方法上回收了众所周知的结果。最后，我们将方法扩展到非欧盟近端几何形状。
We propose an efficient algorithm for finding first-order Nash equilibria in min-max problems of the form $\min_{x \in X}\max_{y\in Y} F(x,y)$, where the objective function is smooth in both variables and concave with respect to $y$; the sets $X$ and $Y$ are convex and "projection-friendly," and $Y$ is compact. Our goal is to find an $(\varepsilon_x,\varepsilon_y)$-first-order Nash equilibrium with respect to a stationarity criterion that is stronger than the commonly used proximal gradient norm. The proposed approach is fairly simple: we perform approximate proximal-point iterations on the primal function, with inexact oracle provided by Nesterov's algorithm run on the regularized function $F(x_t,\cdot)$, $x_t$ being the current primal iterate. The resulting iteration complexity is $O(\varepsilon_x{}^{-2} \varepsilon_y{}^{-1/2})$ up to a logarithmic factor. As a byproduct, the choice $\varepsilon_y = O(\varepsilon_x{}^2)$ allows for the $O(\varepsilon_x{}^{-3})$ complexity of finding an $\varepsilon_x$-stationary point for the standard Moreau envelope of the primal function. Moreover, when the objective is strongly concave with respect to $y$, the complexity estimate for our algorithm improves to $O(\varepsilon_x{}^{-2}{κ_y}^{1/2})$ up to a logarithmic factor, where $κ_y$ is the condition number appropriately adjusted for coupling. In both scenarios, the complexity estimates are the best known so far, and are only known for the (weaker) proximal gradient norm criterion. Meanwhile, our approach is "user-friendly:" (i) the algorithm is built upon running a variant of Nesterov's accelerated algorithm as subroutine and avoids extragradient steps; (ii) the convergence analysis recycles the well-known results on accelerated methods with inexact oracle. Finally, we extend the approach to non-Euclidean proximal geometries.
